1. Field of the Invention
The present invention relates to a profile simulation method for predicting a processed profile obtained by development, etching, or the like, and to a mask design method and a mask design apparatus for designing an exposure mask with use of the simulation method.
2. Description of the Related Art
In a case of manufacturing a semiconductor element and the like by lithography steps, an image of a mask pattern is projected onto a wafer coated with a resist, and thereafter, the wafer is subjected to development, to form a resist pattern. For example, in a case of a positive tone resist, the resist of the portion where the exposure is great is dissolved by a developer, thereby forming a resist pattern.
The resist pattern obtained after the development varies depending on defocus conditions, exposure conditions (e.g., a numerical aperture, a coherence factor, a light source shape, a pupil filter, and the like), a development time, and a mask pattern.
Therefore, a great number of exposure experiments must be carried out in order to obtain conditions and a mask pattern in a projection optical system used for finishing a desired pattern which has a predetermined depth of focus, from experiments. Hence, it is desirable to carry out a simulation using a computer, to obtain conditions in which the resist profile after development has an optimal condition.
Thus, a method of predicting a resist profile after development has been known as a resist profile simulation. Various methods have been proposed as a conventional resist profile simulation method. In the following, those methods will be briefly explained.
There is a string model in which the profile of a resist is expressed as a sequence of fine line segments in a case of dealing with only two-dimensional profiles, while the profile of a resist is expressed as a sequence of fine area segments in a case of dealing with a three-dimensional profile. In this string model, the direction in which fine line segments or area segments are moved is a direction vertical to the surface. In contrast, a method of obtaining a moving direction with use of a differential equation to solve a similar differential equation is called a ray-tracing model.
There is known a cell model as a method in which an object is divided into an aggregate of fine cells and a change in profile is expressed by elimination or adhesion of cells on the surface of the object. In addition, there is a distribution function method in which the profile of an object is expressed as an equivalent area of a distribution function and a differential equation similar to a diffusion equation is solved to obtain a time-based change of the profile.
Further, as a simplified method, there is a simplified development model. In this simplified development model, the development processing is arranged such that an aggregate of end points is the profile after development, supposing that the development proceeds in a direction vertical to the substrate from a start point which is the point on the resist surface where the solution speed is fastest, that the development then proceeds in a direction parallel to the substrate from the point where the development has proceeded for a certain distance, and that the developing direction is changed at all the points vertical to the substrate.
However, these kinds of methods include the following problems. Specifically, in the conventional techniques as described above, the solution speed of resist does not depend on the profile of the portion to be dissolved, but depends on only the photo-sensitive characteristics and processing conditions.
For example, in a case of developing a hole pattern, the amount of resist to be dissolved by a developer for a unit volume in a hole is large, so that the solution speed is extremely reduced. Regardless of the extremely reduced solution speed, the simulation is performed with use of a solution speed equal to a solution speed used for a flat resist profile as supposed in exposure of an entire surface. Therefore, in a hole pattern, the resist profile actually obtained is different from the profile as a simulation result.
In addition, ends of a pattern and a bent portion thereof lead to problems in a case of using an equal solution speed, in addition to the hole pattern. In these portions, the amount of resist to be dissolved by a developer for a unit volume is larger (or smaller) than in the other portions, and the solution speed changes accordingly, so that a simulation result different from an exposure result is obtained.
Meanwhile, a mask pattern is not completely equal to a resist pattern actually finished, and therefore, a mask must be designed estimating the mask to be actually finished, in order to obtain a desired resist pattern. In this case, it is not possible to design a mask with a high accuracy if the actual profile is different from a profile obtained by performing a simulation as described above. Therefore, the procedures are very complicated since it is necessary to actually perform exposure, observe a resist pattern finished by development with use of an electronic microscope, correct the mask pattern in accordance with the observation result, and repeat performing experiments until a desired pattern is obtained.
Thus, in conventional profile simulation methods, there is a problem that a pattern actually obtained is different from a simulation result in the regions of ends and a bent portion of a pattern. In addition, an experiment consisting of a series of exposure, development, observation, and correction must be repeated in order to perform accurate mask designing, and the procedures are thus complicated.